How to add new bone to existing rig?
So let say you've parented your mesh to the armature, With Automatic Weight for example, and that you've created a new bone called arm_L. 

Select the part of the mesh that this bone is supposed to affect, go in the Properties panel > Object Data > Vertex Groups, create a new group called arm_L and click on the Assign button to assign these vertices to this group. If the group has the exact same name as the bone, the bone will automatically influence this vertex group.

Note that on the right of the vertex groups list you have additional functions.

Another method is the Weight Painting, you can't always count on it though.